Orthonormal bases of regular wavelets in spaces of homogeneous type

Adapting the recently developed randomized dyadic structures, we introduce the notion of spline function in geometrically doubling quasi-metric spaces. Such functions have interpolation and reproducing properties as the linear splines in Euclidean spaces. They also have Hölder regularity. This is used to build an orthonormal basis of Hölder-continuous wavelets with exponential decay in any space of homogeneous type. As in the classical theory, wavelet bases provide a universal Calderón reproducing formula to study and develop function space theory and singular integrals. We discuss the examples of Lp spaces, BMO and apply this to a proof of the T(1) theorem. As no extra condition (like ‘reverse doubling’, ‘small boundary’ of balls, etc.) on the space of homogeneous type is required, our results extend a long line of works on the subject.